I have a project where I would like to ring a bell by using a push-pull solenoid as a striker if some logical state is true. The solenoids I have seen are mostly 12V. The Arduino can supply 5V (if I'm not mistaken). Can I use a DC step-up power supply to take the output of a 9V battery to store a 12V charge? If yes, any recommended components and if no, recommendations to accomplish the same. Thanks!
The Arduino is NOT a power source except for sensors using a few mA. Those 9V smoke detector batteries are fairly weak, so try it by itself first. I expect you will need a 12VDC wall wart or a beefy power supply. 12VDC needs 8 to 10 dry cells, or 4 Lithium 18650's and a buck converter.
The Arduino 5V output is very limited and intended only for a couple of low power sensors or LEDs. It cannot be used to power motors, servos, solenoids or bells.
9V block batteries are also suitable only for low power devices. You will almost certainly need an external power supply for the bell, and a relay or logic level MOSFET transistor to active the bell.
Example circuit for motor, but would work with a bell. The kickback diode D1 is required to avoid destroying the MOSFET.
What are the power requirements for the bell? What you have seen is probably not relevant to what you have. I do not have a clue as to what you have seen but I have seen bells with large motors in tall towers and small ones in a phone that like 90 volts. Is it AC or DC?
Use a 12V battery
The bell type is what they call a "call bell". Think of a 4-5" bell on the desk at a hotel, so this is a tiny bell not a large church bell. The solenoid I was looking at has the following specs:
Model: Solenoid Coil--wiring connecting; Voltage: DC12V; Resistance: 1.5Ω; Product size:125×40×35mm/4.92×1.57×1.38"(L×W×H); Core length: 125mm; Stroke: 35mm/1.38"; Weight: 350g
For this application it would need to ring, at most, once per 5 seconds for short periods. A wall wart is not an option as this is to be a mobile device. 18650 batteries are ok and one that I have on hand rates 63W, 22.1A and 3.6V. Would that work? Thanks!
Depending on how loud you need the bell to ring, you can change the solenoid for something a little simpler, like this small DC motor:
https://littlebirdelectronics.com.au/products/micro-mini-strong-vibration-motor
18650 cells could work with a 12V boost converter that can deliver the required current. 12V/1.5R = 8 Amperes or 96 W maximum, which is not at all trivial.
What is your plan to recharge the cells?
That makes no sense.
What you need to know is the maximum current the battery can deliver and the mAh rating.
Max amps is 31.5 and mAH is 3015
That should be OK provided you can find a high current 12V boost.
How long the battery will last will depend on how often you ring the bell.
The cells would be replaceable and rechargeable in an external charger...
Thank you!
So put 3 cells in series and you get 4x battery life without hassle of boost converter..
Thank you! 
Where are you going to strike the bell and where will you solenoid be firmly mounted?
Bell struck from the interior and yes, the solenoid would be screwed down to a board. 